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Let $\mathbb{N}^{2}_{\leqslant}$ be the set $\mathbb{N}^{2}$ with the partialorder defined as the product of usual order $\leq$ on the set of positiveintegers $\mathbb{N}$. We study the semigroup$\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant})$ of monotone injectivepartial selfmaps of $\mathbb{N}^{2}_{\leqslant}$ having cofinite domain andimage. We describe the natural partial order on$\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant})$ and show that itcoincides with the natural partial order which is induced from symmetricinverse monoid $\mathscr{I}_{\mathbb{N}\times\mathbb{N}}$ onto$\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant})$. We proved that thesemigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant})$ is isomorphicto the semidirect product$\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^2_{\leqslant})\rtimes \mathbb{Z}_2$of the monoid $\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^2_{\leqslant})$ oforientation-preserving monotone injective partial selfmaps of$\mathbb{N}^{2}_{\leqslant}$ with cofinite domains and images by the cyclicgroup $\mathbb{Z}_2$ of the order two. Also we describe the congruence $\sigma$on $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant})$ which is generated bythe natural order $\preccurlyeq$ on the semigroup$\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant})$. We prove that thequotient semigroup$\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^2_{\leqslant})/\sigma$ isisomorphic to the free commutative monoid $\mathfrak{AM}_\omega$ over aninfinite countable set and show that the quotient semigroup$\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant})/\sigma$ is isomorphic tothe semidirect product of the free commutative monoid $\mathfrak{AM}_\omega$ by$\mathbb{Z}_2$.

On the monoid of monotone injective partial selfmaps of $\mathbb{N}^{2}_{\leqslant}$ with cofinite domains and images, II